Contractions with Polynomial characteristic functions I. Geometric approach

In this note we study the completely non unitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form T = S & * & * 0 & N & * 0& 0& C, where $S$ and C^* are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that dimension of ker S^* and dimension of ker C are unitary invariants of $T$ and that N, up to a quasi-similarity is uniquely determined by T. Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.